Seminars and Colloquia by Series

Mechanisms of Chaos

Series
Research Horizons Seminar
Time
Wednesday, October 28, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Leonid Bunimovich – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided before the seminar

In this seminar,we will explain why and how unpredictable (chaotic) dynamics arises in deterministic systems. Some open problems in dynamical systems, probability, statistical mechanics, optics, (differential) geometry and number theory will be formulated.

Generalized Dantzig Selector: Application to the k-support norm

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, October 27, 2015 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 249
Speaker
Changong Li – Georgia Inst. of Technology, School of Mathematics

Please Note: Review of a recent paper by Chatterjee et al. (Arxiv 1406.5291)

We propose a Generalized Dantzig Selector (GDS) for linear models, in which any norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate proximal operator, a flexible inexact ADMM framework is designed for solving GDS, and non-asymptotic high-probability bounds are established on the estimation error, which rely on Gaussian width of unit norm ball and suitable set encompassing estimation error. Further, we consider a non-trivial example of the GDS using k-support norm. We derive an efficient method to compute the proximal operator for k-support norm since existing methods are inapplicable in this setting. For statistical analysis, we provide upper bounds for the Gaussian widths needed in the GDS analysis, yielding the first statistical recovery guarantee for estimation with the k-support norm. The experimental results confirm our theoretical analysis.

Relative Entropy Relaxations for Signomial Optimization

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, October 27, 2015 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Venkat Chandrasekaran – Cal Tech
Due to its favorable analytical properties, the relative entropy function plays a prominent role in a variety of contexts in information theory and in statistics. In this talk, I'll discuss some of the beneficial computational properties of this function by describing a class of relative-entropy-based convex relaxations for obtaining bounds on signomials programs (SPs), which arise commonly in many problems domains. SPs are non-convex in general, and families of NP-hard problems can be reduced to SPs. By appealing to representation theorems from real algebraic geometry, we show that sequences of bounds obtained by solving increasingly larger relative entropy programs converge to the global optima for broad classes of SPs. The central idea underlying our approach is a connection between the relative entropy function and efficient proofs of nonnegativity via the arithmetic-geometric-mean inequality. (Joint work with Parikshit Shah.)

Seismic inverse problems

Series
IMPACT Distinguished Lecture
Time
Tuesday, October 27, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Maarten de Hoop – Rice University –
We give a brief analysis of the oscillations of the earth and then extract the system of equations describing acousto-elastic, seismic waves. Processes in Earth's interior are encoded in the coefficients of this system, which also parametrize its structure and material properties. We introduce the seismic inverse problem with its different aspects including a dual time-frequency point of view. Central in the analysis is the formulation as an inverse boundary value problem with the Dirichlet-to-Neumann map or Neumann-to-Dirichlet map as the data. We discuss various conditional Lipschitz stability estimates for this problem for coefficients containing discontinuities, and with partial boundary data, which involves the introduction of an unstructured tetrahedral mesh. Quantitative estimates of the stability constants play acritical role in analyzing convergence for iterative reconstruction schemes, making use of Hausdorff warping and leading to a multilevel approach requiring hierarchical, multi-scale compression. We present computational experiments on the regional and geophysical exploration scales. We conclude with some results pertaining to the high-frequency inverse boundary value or geometric inverse problems, again, in the presence of discontinuities.

Repairing tropical curves by means of tropical modifications

Series
Algebra Seminar
Time
Monday, October 26, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Maria Angelica Cueto – The Ohio State University
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. In this talk, I will advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input one. Our motivating examples will be plane elliptic cubics and genus two hyperelliptic curves. Based on joint work with Hannah Markwig (arXiv:1409.7430) and ongoing work in progress with Hannah Markwig and Ralph Morrison.

Triangulation independent Ptolemy varieties

Series
Geometry Topology Seminar
Time
Monday, October 26, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Christian Zickert – University of Maryland –
The Ptolemy variety is an invariant of a triangulated 3-manifoldM. It detects SL(2,C)-representations of pi_1(M) in the sense that everypoint in the Ptolemy variety canonically determines a representation (up toconjugation). It is closely related to Thurston's gluing equation varietyfor PSL(2,C)-representations. Unfortunately, both the gluing equationvariety and the Ptolemy variety depend on the triangulation and may missseveral components of representations. We discuss the basic properties ofthese varieties, how to compute invariants such as trace fields and complexvolume, and how to obtain a variety, which is independent of thetriangulation.

Uniqueness of seismic inverse source problems modeling microseismicity and ruptures

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 26, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Maarten de Hoop – Rice University –
We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along an unknown path with subsonic velocity, and that data is collected over time on some detection surface. We explore the question of uniqueness for these problems, and show how to recover the times and locations of sources microlocally first, and then the smooth part of the source assuming that it is the same at each source location. In case the sources (now all different) are (roughly speaking) non-negative and of limited oscillation in space, and sufficiently separated in space-time, which is a model for microseismicity, we present an explicit reconstruction, requiring sufficient local energy decay. (Joint research with L. Oksanen and J. Tittelfitz)

Calculation of a Power Price Equilibrium under Risk Averse Trading

Series
Other Talks
Time
Monday, October 26, 2015 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Raphael Hauser – Mathematical Institute, University of Oxford
We propose a term structure power price model that, in contrast to widely accepted no-arbitrage based approaches, accounts for the non-storable nature of power. It belongs to a class of equilibrium game theoretic models with players divided into producers and consumers. The consumers' goal is to maximize a mean-variance utility function subject to satisfying an inelastic demand of their own clients (e.g households, businesses etc.) to whom they sell the power. The producers, who own a portfolio of power plants each defined by a running fuel (e.g. gas, coal, oil...) and physical characteristics (e.g. efficiency, capacity, ramp up/down times...), similarly, seek to maximize a mean-variance utility function consisting of power, fuel, and emission prices subject to production constraints. Our goal is to determine the term structure of the power price at which production matches consumption. We show that in such a setting the equilibrium price exists and discuss the conditions for its uniqueness. The model is then extended to account for transaction costs and liquidity considerations in actual trading. Our numerical simulations examine the properties of the term structure and its dependence on various model parameters. We then further extend the model to account for the startup costs of power plants. In contrast to other approaches presented in the literature, we incorporate the startup costs in a mathematically rigorous manner without relying on ad hoc heuristics. Through numerical simulations applied to the entire UK power grid, we demonstrate that the inclusion of startup costs is necessary for the modeling of electricity prices in realistic power systems. Numerical results show that startup costs make electricity prices very spiky. In a final refinement of the model, we include a grid operator responsible for managing the grid. Numerical simulations demonstrate that robust decision making of the grid operator can significantly decrease the number and severity of spikes in the electricity price and improve the reliability of the power grid.

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